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230 IX ANALYTIC FUNCTIONS
(b) Give an example of an indefinitely differentiable function in R which is not analytic
(cf. Section 8.12, Problem 2).
(c) Suppose/is real valued and indefinitely differentiable in an open interval I c R;
in addition, suppose that there is an integer p ^ 0 such that /(ll) does not vanish at more than p points of I, for any n > 0. Show that / is analytic in I. (Use (a), and Problem 3(b) of Section 8.12). |
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10. CHARACTERIZATION OF ANALYTIC FUNCTIONS OF
COMPLEX VARIABLES
(9.10.1) A continuously differentiable mapping f of an open subset A of Cp
into a complex Banach space is analytic.
Applying (9.9.4), we are immediately reduced to the case p = l.
To prove / is analytic at a point a e A, we may, by translation and homothetic mapping, suppose that a = 0 and that A contains the unit
o
ball B: \z\ ^ 1. For any ze B, and any A such that 0 < 1 ^ 1, note that
|(1 — A)z + lelt | < 1 — A + 1 = I, and consider the integral |
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e z
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By (8.11.1) and Leibniz's rule (8.11.2), g is continuous in [0, 1] and has at
each point of ]0, 1[ a derivative equal to |
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•-£
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(see Remarks after (8.4.1)). But ikf'(z + A(eu - z))elt is the derivative of
/ -»/(z -f A(eif - z)), hence, for A ^ 0, g'(X) = 0, and therefore (remark following (8.6.1)),#isco/wtaflnn[0, l].Butas#(0) = 0,#(A) = OforO ^ A ^ 1. In particular, it follows, for A = 1, that
f(x)dx
for any z G B (by (9.8.4)), and the conclusion follows from (9.9.2).
(9.10.2) Let f be a continuously differentiable mapping of an open set
A c: R2p into a complex Banach space. In order that the function g defined in A (considered as a subset of Cp), by f(xl9 x2, ..., xp, yl9... ,yp) = g (#1 + z>i,.. -, xp H- z>p) be analytic in A, necessary and sufficient conditions are that |
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dxk dyk
in A for 1 < k ^p (Cauchy's conditions). |
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